Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910335486656512 |
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| author | Kuperberg, Vivian Lalín, Matilde |
| author_facet | Kuperberg, Vivian Lalín, Matilde |
| contents | In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_k(f)$ over the function field $\mathbb{F}_q[T]$ in the limit as $q \to \infty$ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [arXiv:1504.07804] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04969 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime Kuperberg, Vivian Lalín, Matilde Number Theory 11M50, 11T55 In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_k(f)$ over the function field $\mathbb{F}_q[T]$ in the limit as $q \to \infty$ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [arXiv:1504.07804] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields. |
| title | Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime |
| topic | Number Theory 11M50, 11T55 |
| url | https://arxiv.org/abs/2212.04969 |